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A benefit of sexual reproduction

This document aims to answer the question "What is the advantage of sexual reproduction, that compensates for the twofold disadvantage?


History of the problem

Consider a female who will give birth to n progeny. If she can choose between producing n progeny identical to herself and n each carrying half her genes and half the genes provided by a male, the former seems clearly more in her genetic interests. Yet most higher animals and plants, including almost all mammals, use sexual reproduction every generation. This needs explaining.

Many explanations have been offered, particularly in the 1970s. These included the "aphid rotifer", "strawberry-coral", and "elm-oyster" models[A], the "tangled bank" idea[B], the "libertine bubble" idea[C], and the "Red Queen hypothesis"[D]. More recent ideas have been the counteracting of Muller's ratchet[E], the "removal of minor faults"[F], the possibility of providing an unbounded rate of information gain[G], and its rôle in the host-parasite arms race[H].

Most studies, both experimental and theoretical, have concentrated on relatively high selective pressures. For experiments, this is clearly necessary. We concentrate on low selective pressures, at many loci, using a computer model described below. We believe this clarifies a major benefit of sexual reproduction, in a way which is compatible with many of the hypotheses listed above.

Description of the model


Explanation of the graphs


Interpretation of the graphs


Above is a sample graph, for a haploid population. The population size is 2000, the mutation rate is 0.001 per individual per generation, and the finesses of mutant alleles is rectangularly distributed between 0 and 1.1 (so most are disadvantageous).

The first four alleles to achieve a frequency of over 0.1 all rise smoothly to fixation, in sigmoid shapes. The first of these becomes fixed in about generation 250, to the left of the green line, so is not inluded in the total "Fixed 7".

The fifth allele to rise above 0.1 frequency peaks at about 0.14, and then rapidly falls back to 0 because it is competing with a new and fitter allele which appeared at another locus in generation 600. Then that new and fitter allele itself starts to fall back, because of competition from two more new alleles. But after falling to a frequency just below 0.6, it starts to rise again. This is because yet another fit new allele, arising in an individual which already had the "generation 600" allele, helps it to compete. (We can tell that it did so because the graphs for the two alleles eventually merge at generation 920: thereafter all the individuals that have the generation-600 allele also have its accomplice).

When looking at a graph for an asexual population, we often see two lines merge into a single line. This happens when the more recent and rarer of the two alleles has appeared in an individual already having the older allele. Also, with an asexual population, we often see two lines "mirror" each other: wherever one rises, the other falls by the same amount, as at generations 700 to 900 in the graph above. This happens when the entire population carries one or another of the alleles, but not both. The two lines continue to rise and fall, not only because of drift, but because of the appearance of other new alleles within one or the other of the two subpopulations.

The paragraph above is italicised because it is a key to understanding these graphs. The merging, and to some extent the mirroring, are also seen in graphs for sexual populations, but are less prevalent there.

Above is a graph with the same parameters as the previous one, except that it is for a sexual population. It shows similar sigmoid ascents, but with the difference that rather than "competing", two of the sigmoids can continue upwards together. We see a low-slope sigmoid, that achieves 0.5 frequency in generation 325, continue its slow upwards progress while being overtaken by other newer and fitter alleles. Unlike in the haploid case, the appearance of fitter alleles at other loci is not a setback for it, instead, individuals carrying both beneficial alleles are generated by sex.

Some results

1. Haploid compared with sexual haploid

Using pairs of graphs such as those shown in the previous section, but run for more generations, we see that new alleles achieve fixation at a higher rate in a "sexual haploid" population than in a haploid population.

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2. Panmictic compared with unmixed

If we compare a panmictic sexual haploid population with a sexual haploid population without mixing (the individuals are treated as being in a line, each pairs with one of the adjacent individuals), we find that the former has new alleles achieving fixation at a higher rate. Both do better than a haploid population (for which mixing has no effect).

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3. With and without adverse mutations

We compare the effect of having adverse mutations (the fitness of a new allele is rectangular in the range 0 to 1.1) and having no adverse mutations (the fitness of a new allele is rectangular in the range 1 to 1.1, and the total mutation rate is 1/11 as much). We find that there is no significant difference in the rate of fixation of new alleles, in both sexual haploid and haploid simulated populations. However, the variance in the fitness of evolving populations is higher if there are adverse mutations.

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4. The effect of population size

In both haploid and sexual haploid populations, the rate of fixation of new alleles is higher in larger populations. This effect of population size is greater in sexual haploid than in haploid populations.

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5. The effect of a finite number of loci

In both haploid and sexual haploid populations, the rate of fixation of new alleles is higher in larger populations. This effect of population size is greater in sexual haploid than in haploid populations.

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We see that effects described above with an infinite number of loci are still seen when the number is finite.

In the lower half of the graphs in this section, with the fitnesses of new alleles randomly chosen in the range 0 – 1.1, most of these alleles will be deleterious and very rarely achieve fixation. So the number that are ever likely to be fixed is limited to approximately 1/11 of the number of loci.

6. The efficiency of low selection rates

We compare the effect of fitnesses of new alleles in the range 1 – x, with x varying from 0.1 to 0.00001. For each value, over sexual haploid populations of size 1000 with infinite loci, we examine the number of new alleles fixed and the cumulative log variance of the fitness. The latter is a measure of the cumulated rate of selective deaths in the evolving population.

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Thus we have, for the conditions of these simulations:
    No. of alleles fixed is proportional to x0.9
    Total variance in fitness is proportional to x1.9

7. Finite and infinite loci

Do these simulation for an infinite number of loci give the same results as for a number large enough not to "run out" in the course of the simulation?

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We see that the simulations with infinite loci have very similar results to those with infinite loci. So our code treats "an infinite number of loci" in the same way as "a very large number of loci".

We also see that for the lowest mutation rate, 0.05 mutations per individual per generation, the fixation rate is higher for haploids than for sexual haploids. This may be because with sexual haploids, each new allele spreads very slowly, not much faster than it would by drift, and very few are fixed in the course of this simulation; whereas in a haploid population, several beneficial mutations that happen to occur in the same bloodline can rise together at a significantly faster rate than drift, without being separated by sexual reproduction.


From the comparisons above, specifically Haploid compared with sexual haploid and The effect of population size, we conclude that

Acquiring a new allele very slowly is very efficient in selective deaths.
Acquiring mutliple new alleles at once, all very slowly, is very efficient only if the acquisitions do not interact negatively with each other, as they do in asexual populations. Sexual reproduction reduces negative interactions. Frequent sexual reproduction combined with panmixis prevents negative interactions.

Further questions to consider



George C. Williams. 1975. Sex and Evolution, Princeton University Press

Graham Bell. 1982. The Evolution and Genetics of Sexuality. University of California Press, Berkeley.

Thierry Lodé. 2011. Sex is not a good solution for reproduction: the libertine bubble theory. Bioessay 33: 419–432

Leigh Van Valen. 1973. "A new evolutionary law". Evolutionary Theory. 1: 1–30.

Hermann Joseph Muller. 1932. "Some genetic aspects of sex". American Naturalist. 66 (703): 118–138

Alexey Kondrashov. 1988. "Deleterious mutations and the evolution of sexual reproduction". Nature 336: 435-440

David J.C. MacKay. 2003. Information Theory, Inference, and Learning Algorithms, C.U.P. 2003: chapter 19.
"An information theoretic analysis using a simplified but useful model shows that in asexual reproduction, the information gain per generation of a species is limited to 1 bit per generation, while in sexual reproduction, the information gain is bounded by sqrt(G}, where G is the size of the genome in bits."

Lively & Morran. 2014. The Ecology of Sexual Reproduction. Journal of Evolutionary Biology, 1292-1303
doi: 10.1111/jeb.12354

Copyright N.S.Wedd 2018.